 12.1: 4
 12.2: 45
 12.3: 6. 23
 12.4: 2 3 _3 2 = 1 5
 12.5: (a + 4) + 2 = a + (4 + 2)
 12.6: 4x + 0 = 4x
 12.7: 8
 12.8: 1 3
 12.9: 1.5
 12.10: Write an expression to represent the total amount of money Catalina...
 12.11: Evaluate the expression from Exercise 10 by using the Distributive ...
 12.12: 3(5c + 4d) + 6(d  2c)
 12.13: 1 2 (16  4a)  _3 4 (12 + 20a)
 12.14: _2 9
 12.15: 4.55
 12.16: 10
 12.17: 19
 12.18: 31
 12.19: 12 2
 12.20: 121 21
 12.21: 36
 12.22: 5a + (5a) = 0 2
 12.23: 6xy + 0 = 6xy
 12.24: [5 + (2)] + (4) = 5 + [2 + (4)] 2
 12.25: (2 + 14) + 3 = 3 + (2 + 14)
 12.26: (1 _2 7)(_7 9) = 1
 12.27: 2 3 + 5 3 = (2 + 5) 3 Ide
 12.28: 10 2
 12.29: 2.5
 12.30: 0.125
 12.31: _5 8
 12.32: 4 3
 12.33: 4 _3 5
 12.34: BASKETBALL Illustrate the Distributive Property by writing two expr...
 12.35: BAKING Mitena is making two types of cookies. The first recipe call...
 12.36: 7a + 3b  4a  5b
 12.37: 3x + 5y + 7x  3y
 12.38: 3(15x  9y) + 5(4y  x)
 12.39: 2(10m  7a) + 3(8a  3m)
 12.40: 8(r + 7t)  4(13t + 5r)
 12.41: 4(14c  10d)  6(d + 4c)
 12.42: 4(0.2m  0.3n)  6(0.7m  0.5n)
 12.43: 7(0.2p + 0.3q) + 5(0.6p  q)
 12.44: If Andrea earns $6.50 an hour, illustrate the Distributive Property...
 12.45: Find the mean or average number of hours Andrea worked each day, to...
 12.46: If m + n = m, what is the value of n?
 12.47: If m + n = 0, what is the value of n? What is ns relationship to m?
 12.48: If mn = 1, what is the value of n? What is ns relationship to m?
 12.49: If mn = m and m 0, what is the value of n?
 12.50: To what set of numbers was Pythagoras referring when he spoke of nu...
 12.51: Use the formula c = 2s2 to calculate the length of the hypotenuse c...
 12.52: Explain why Pythagoras could not find a number for the value of c.
 12.53: 0
 12.54: 3 2
 12.55: 2 7
 12.56: Name the sets of numbers to which all of the following numbers belo...
 12.57: integer, but not a natural number
 12.58: integer with a multiplicative inverse that is an integer
 12.59: Every whole number is an integer. 6
 12.60: Every integer is a whole number.
 12.61: Every real number is irrational. 6
 12.62: Every integer is a rational number.
 12.63: REASONING Is the Distributive Property also true for division? In o...
 12.64: Writing in Math Use the information about coupons on page 11 to exp...
 12.65: ACT/SAT If a and b are natural numbers, then which of the following...
 12.66: REVIEW Which equation is equivalent to 4(9  3x) = 7  2(6  5x)?
 12.67: 9(4  3)5
 12.68: 5 + 9 3(3)  8
 12.69: a + 2b  c
 12.70: b + 3(a + d) 3
 12.71: GEOMETRY The formula for the surface area SA of a rectangular prism...
 12.72: 8b  5 7
 12.73: 2 5 b + 1
 12.74: 1.5c  7
 12.75: 9(a  6)
Solutions for Chapter 12: Properties of Real Numbers
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 12: Properties of Real Numbers
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12: Properties of Real Numbers includes 75 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Since 75 problems in chapter 12: Properties of Real Numbers have been answered, more than 26074 students have viewed full stepbystep solutions from this chapter.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.